> I am not a cosmologist, but if all of space reached thermal equilibrium, if space does have any mass (perhaps a tiny amount from the dark energy it contains?), and the entire universe is essentially one uniform mass, wouldn't that mass then collapse upon itself, in a kind of big bounce?

Nor am I, but as long as the universe keeps expanding and everything keeps flying apart the average density will probably continue to decrease and gravity will be too weak to cause a collapse.

Or are you saying dark energy (intrinsic to space itself) would eventually dominate and concentrate?